#166699

order	512
nilpotency class	3
isoclinism	isoclinic class of #166289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#38487	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(3,4)(7,8)(13,16)(17,18,22,26)(19,27,28,31)(20,25,21,24)(23,30,32,29);(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(2,7)(6,8)(19,28)(23,32)(27,31)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73895);